R E S E A R C H
Open Access
Structure of positive solution sets of
differential boundary value problems
Xu Xian and Zhu Xunxia
**Correspondence:
[email protected] Department of Mathematics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, P.R. China
Abstract
In this paper, we first obtain some results on the structure of positive solution sets of differential boundary value problems. Then by using the results, we obtain an existence result for differential boundary value problems. The method used to show the main result is the global bifurcation theory.
Keywords: structure of positive solution sets; differential boundary value problems; bifurcation theory
1 Introduction
This paper considers the differential boundary value problem ⎧
⎨ ⎩
(p(t)φ(u))+λp(t)f(t,u) = , a<t<b,
u(a) =u(b) = , (.)
wheref isφ-superlinear at∞andf(t, ) maybe negative andpis a positive continuous function,λ> is a parameter.
Equations of form (.) occur in the study for thep-Laplacian equation, non-Newtonial fluid theory and the turbulent flow of a gas in a porous medium. The case where
α|∇u||∇u|=|∇u|p–|∇u|, p> ,
i.e., perturbations of thep-Laplacian, has received much attention in the recent literature. Also, problem (.) withf(t, ) > has been studied by several authors in recent years (see [] and the references therein). Here, we are interested in the case whenf(t, ) may be negative (the so-called semipositone case) (see [] and its references for a review). As pointed out by Lions in [], semi-positone problems are mathematically very challenging. During the last ten years, finding positive solutions to semi-positone problems has been actively pursued and significant progress on semi-positone problems has taken place; see [–] and the references therein. For instance, Haiet al.[] considered the existence pos-itive solution of (.). Under some super-linear conditions on the non-linear termf, they proved that there existsλ∗> such that (.) has one positive solution for <λ<λ∗. The main method in [] used to show the main result are the fixed-point theorems.
The main purpose of this paper is going to study the structure of the positive set of (.). Rabinowitz [] gave the first important results on the structure of the solution sets of
non-linear equations and obtained by the degree theoretic method. Amamn [] studied the structure of the positive solution set of non-linear equations; the reader is referred to [, ] for other results concerning the structure of solution sets of non-linear equations. In our paper, we will study the existence results for an unbounded connected component of a positive solution set for the differential boundary value problem of (.). This paper generalizes some results from the literature []. The paper is arranged as follows. In Sec-tion , we will give some preliminary lemmas. The main results will be given in SecSec-tion .
2 Some lemmas
For convenience, we make the following assumptions:
(A) φis an odd, increasing homeomorphism onRwithφ–concave onR+.
(A) For eachc> , there existsAc> such thatφ–(cu)≥Acφ–(u),u∈R+and
limc→∞Ac= +∞(note that (A) implies the existence ofBc> such that φ–(cu)≤Bcφ–(u),u∈R+andlim
c→∞Bc= ). (A) p: [a,b]→(, +∞)is continuous.
(A) f: [a,b]×R→Ris continuous and
lim
u→+∞
f(t,u)
φ(u) = +∞
uniformly fort∈[a,b].
LetE=C[, ], the usual real Banach space of continuous functions with the maximum norm · . Lete(t) =(t–(ba–)(ab)–t) fort∈[a,b] andP={x∈E:u(t)≥,∀t∈[a,b]}. ThenPis
a cone ofE. Define
f∗(t,u) = ⎧ ⎨ ⎩
f(t,u), u≥, f(t, ), u< .
Thenf∗(t,u)≥–mforu∈R(herem> is a constant). Foru∈E,λ∈R+, define
A(λ,u) = t
a φ–
C p(s)–
λ
p(s)
s
a
p(r)f∗(r,u)dr ds,
hereCis a constant such that
b
a φ–
C p(s)–
λ
p(s) s
a
p(r)f∗(r,u)dr ds= .
We know thatCexists and is unique for everyu∈P(see []). Thenu=A(λ,u) if and only ifuis a solution of
⎧ ⎨ ⎩
(p(t)φ(u))+λp(t)f∗(t,u) = , a<t<b,
u(a) =u(b) = .
Lemma . Letφsatisfy(A)and(A).Then for each d> ,there exist constants Kdand
Ldsuch that
Kdφ(x)≥φ(dx)≥Ldφ(x), ∀x≥.
Further,Ld→as d→and Kd→ ∞as d→ ∞.
Lemma . Letφ be as in Lemma..Then there exist <α≤andβ≥such that
φ–(x–y)≥αφ–(x) –βφ–(y)for x≥,y≥.
Lemma . Let p,p > such that p≤p(t)≤p for t∈[a,b].Letλ> andωbe a solution of
⎧ ⎨ ⎩
(p(t)φ(ω))+λp(t)h(t) = , t∈[a,b],
ω(a) =ω(b) = , (.)
here h(t)is continuous function with h(t)≥–m(here m> is a constant)for t∈[a,b],if
ω ≥β(bα–a)φ–(λmδ),then
ω(t)≥αω–βφ–(λmδ)(b–a)e(t) for t∈[a,b], whereδ=p(b–a)
p .
Proof By integrating, it follows that (.) has the unique solution given by
ω(t) = t a φ– p(s) C–λ
s
a
p(r)h(r)dr ds,
whereCis such thatω(b) = . Letω∞=|ω(t)|for somet∈(a,b). Then
ω(t) = t a φ– λ p(s) t s
p(r)h(r)dr– λm p(s)
t
s
p(r)dr ds,
whereh(t) =h(t) +m≥. By Lemma ., we get
ω(t)≥α
t a φ– λ p(s) t s p(r)h(r)dr
ds–β
t a φ– λm p(s) t s
p(r)dr ds.
Now t a φ– λm p(s) t s
p(r)dr ds≤ t a φ– λm p t s
pdr ds
≤
t
a φ–
λmp p
(b–a) ds
=φ–(λmδ)(t–a).
And so
here
ω(t) = t
a φ–
λ
p(s) t
s
p(r)h(r)dr
ds fort∈[a,t].
Note thatωsatisfies ⎧
⎨ ⎩
(p(t)φ(ω))+λp(t)h(t) = , t∈[a,t],
ω(a) = , ω(t)≥ |ω(t)|.
In fact,ω(t)≥ω(t) fort∈[a,t]. We next prove thatω(t)≥ν(t) fort∈[a,t], hereν satisfies
⎧ ⎨ ⎩
(p(t)φ(ν))= , t∈[a,t],
ν(a) = , ν(t) =ω.
Suppose it is not true, thenω–νhas a negative absolute minimum atτ∈(a,t). Since
ω(a) –ν(a) = , ω(t) –ν(t)≥, there existτ,τ∈[a,t] such that
ω(τ) –ν(τ) =ω(τ) –ν(τ) = ,
and
ω(t) –ν(t) < , t∈(τ,τ). Then
p(t)φω(t)–p(t)φν(t)= –λp(t)h(t)≤ fort∈(τ,τ). Letu(t) =ω(t) –ν(t),t∈(τ,τ), then
τ
τ
p(t)φω(t)–p(t)φν(t)u(t)dt≥.
On the other hand, using the inequality
φ(b) –φ(a)(b–a)≥, a,b∈R,
and the fact that there existsτ∗∈[τ,τ] such thatω(τ∗)=ν(τ∗), we have
τ
τ
p(t)φω(t)–p(t)φν(t)u(t)dt
= – τ
τ
which is a contradiction. So,ω(t)≥ν(t) fort∈[a,t]. Obviously,ν(t) = tω–a(t–a),t∈ [a,t], sinceω≥θfor eacht∈[a,t]. From (.), we have
ω(t)≥αω–βφ–(λmδ)(b–a)t–a t–a
, t∈[a,t].
Similarly,
ω(t)≥αω–βφ–(λmδ)(b–a)b–t b–t
, t∈[t,b].
Ifω ≥ β(bα–a)φ–(λmδ), then
ω(t)≥αω–βφ–(λmδ)(b–a)e(t) fort∈[a,b].
The proof is complete.
LetQλ={u∈E|u≥(αu–βφ–(λmδ)(b–a))e(t),∀t∈[a,b]}for eachλ∈R+, where α,β> andδ> are defined as that in Lemma . and Lemma ., respectively. ThenQλ
is also a cone ofE. From Lemma ., we know thatA:P→Qλis completely continuous.
Let
L(P) =(λ,u)|λ∈R+,u∈P\{θ}is a solution of (.). From [, Lemma .], we have Lemma ..
Lemma . Let X be a compact metric space.Assume that A and B are two disjoint closed subsets of X.Then either there exist a connected component of X meeting both A and B or X=A∪BwhereA,Bare disjoint compact subsets of X containing A and B,
respec-tively.
Let U be an open and bounded subset of the metric space[a,b]×E.We set U(λ) ={x∈ E: (λ,x)∈U},whose boundary is denoted by∂U(λ).Consider a map h(λ,x) =x–k(λ,x), such that k(λ,·)is compact andθ∈/h(∂U).Such a map h will also be called an admissible homotopy on U.If h is an admissible homotopy,for everyλ∈[a,b]and every x∈∂U(λ), one has that hλ(x) :=h(λ,x)=θand it makes sense to evaluatedeg(hλ,U(λ),θ).
Lemma . If h is an admissible homotopy on U⊂[a,b]×E,thedeg(hλ,U(λ),θ)is
con-stant for allλ∈[a,b].
Lemma . Let h∈E\{θ}such that h≥αhe(t).Then for arbitraryλ∈(, ],there exists Rλ> such that for eachλ∈[λ, ],R≥Rλandμ≥,
u=Aλ,u+μh, ∀u∈∂Bθ,R, where B(θ,R) ={u:u<R}.
Proof From (A), form> , such that
α(b–a)
φ
–
mλp(b–a) p
there existsm> such thatf∗(t,u)≥mφ(u) foru≥m. Let
Rλ=max
α
m mint∈[b+a
,b+a]e(t)
+βφ–λmδ(b–a)
,β(b–a)
α φ
–(λ mδ)
+ .
Assume by contradiction that
u=A(λ,u) +μh (.)
for someR≥Rλ,u∈∂B(θ,R),λ∈[λ, ] andμ≥. Let
y(t) =A(λ,u) =
t a φ– C p(s)– λ p(s) s a
p(r)f∗(r,u)dr ds.
From Lemma ., we know thaty(t)∈Qλ. Namely,
y(t)≥
αy–βφ–(λmδ)(b–a)
e(t). (.)
From (.) and (.), we have
u =y(t) +μh
≥αy–βφ–
λmδ(b–a)
e(t) +μαhe(t)
≥(αy+μh–βφ–
λmδ(b–a)
e(t)
=αu–βφ–(λmδ)(b–a)
e(t),
sou∈Qλ. Forλ∈[λ, ], we have
u(t)≥
αu–βφ–
λmδ(b–a)
e(t)≥m, fort∈(a+b
,
a+b
). Therefore, letu=u(tu),tu∈(
a+b
,
a+b
), assume thattu≥
a+b
, then
u=u(tu)
= tu
a+b
φ– λ p(s) tu s
p(τ)f∗(t,u)dτ ds+μh
≥α
tu
a+b
φ–
λp p
tu
s
p(τ)f∗(t,u) +m
dτ ds
–β
tu
a+b
φ–
λpm p
(tu–s) ds,
where tu
a+b
φ–
λp p
tu
s
f∗(τ,u) +m
dτ ds
≥
a+b
a+b
φ–
λp p
tu
s
f∗(τ,u) +m
≥
a+b
a+b
φ–
λp p
a+b
a+b
mφ
u
dτ ds
= a+b
a+b
φ–
mλp(b–a) p
uds
=b–a φ
–
mλp(b–a)
p
u,
and tu
a+b
φ–
λpm p
(tu–s) ds
≤
tu
a+b
φ–
λpm p
(b–a) ds
=φ–
λpm p
(b–a) tu–
a+b
≤(b–a)
φ
–(λ mδ).
Thus,
u ≥
α(b–a)
φ
–
mλp(b–a)
p
u–
β(b–a)
φ
–(λ mδ)
≥α(b–a)
φ
–
mλp(b–a)
p
u–
α(b–a)
u
=
α(b–a)
φ
–
mλp(b–a) p
–α(b–a)
u,
so α(b–a)φ–(mλp(b–a)
p ) –
α(b–a)
≤, which is a contradiction. Then (.) holds. The proof
is complete.
3 Main results
For convenience, let us introduce the following symbols. For anyr,
M(r) =(λ,u)∈[, ]×E:u=r,
M[r, +∞) =(λ,u)∈[, ]×E:r≤ u< +∞, M[,r) =(λ,u)∈[, ]×E: ≤ u<r.
Now we give our main results of this paper.
Theorem . Suppose(A)to(A)hold.Then L(P)∩([, ]×P)possesses an unbounded connected component D∗such thatProjλD∗⊃(,λ∗]for someλ∗> and
lim
λ→+,(λ,u)∈D∗u= +∞,
Proof We divide our proof into four steps. Step . Let
T(λ,u) = ⎧ ⎨ ⎩
A(λ,u), ([, ]×E)∩M[, +∞),
θ, ([, ]×B(θ, ))∪({} ×E), (.)
whereB(θ, ) ={u∈E:u ≤}. Obviously,T(·,·) is a completely continuous operator. Note thatT(λ,u) =A(λ,u) for allλ∈[, ] andu∈Ewithu ≥. From Lemma ., there existsR> large enough such thatFixT(,·)⊂B(θ,R) and
degI–T(,·),B(θ,R),θ
= . (.)
Obviously,
degI–T(,·),B(θ, ),θ= . (.)
Therefore,
degI–T(,·),B(θ,R)\B(θ, ),θ
=degI–T(,·),B(θ,R),θ
–degI–T(,·),B(θ, ),θ
= – = –. (.)
So,FixT(,·)=∅andFixT(,·)⊂U:=B(θ,R)\B(θ, ). Step . Let
S=(λ,u)∈R+×E:u=T(λ,u). From Lemma ., we have
L(P)∩[, ]×u∈E: <u ≤=∅.
This implies thatThas no bifurcation point on [, ]. From step ,we haveL(P)∩({}×E)=
∅, then for each (,u)∈L(P)∩({} ×E), denote byDuthe connected component of the
metric spaceL(P)∩({} ×E) emitting from (,u). Now we will show that, there must exist (,u)∈L(P)∩({} ×E) such thatDu is unbounded. Assume on the contrary thatDu
is bounded for each (,u)∈L(P)∩({} ×E). Take a bounded open neighborhoodU
uin
L(P)∩({} ×E) for eachDusuch that
Cl({}×E)Uu∩
[, ]× {θ}=∅ (.)
andU
u∩({} ×E) =∅, whereCl({}×E)Uu denotes the closure ofUu in the metric space
[, ]×E. Let∂[,]×EUu denote the boundary ofUu in the metric space [, ]×E.
Obvi-ously,∂[,]×EUu∩L(P) is a compact subset. Assume that∂[,]×EUu∩L(P)=∅. From the
maximal connectedness ofDu, there is no connected subset of∂[,]×EUu∩L(P)
()u and()u ofCl[,]×E()u ∩L(P) such thatDu⊂u()and∂[,]×EUu()∩L(P)⊂()u , and
Cl[,]×EUu()∩L(P) =()u ∪()u . Letd=d(()u ,()u ) > andUu()be thed-neighborhood
of()u in the metric space [, ]×E. Set
Uu=
⎧ ⎨ ⎩
Uu()∩Uu(), when∂[,]×EUu()∩L(P)=∅,
Uu(), when∂[,]×EUu()∩L(P) =∅.
(.)
Then we haveDu⊂Uuand
∂[,]×EUu∩L(P) =∅. (.)
Obviously, the collection of the subsets
Uu∩ {} ×E: (,u)∈L(P)∩
{} ×E
is an open cover ofL(P)∩({} ×E). SinceL(P)∩({} ×E) is compact, then there exist finite points, namely
(,u), (,u), . . . , (,un)∈
{} ×E∩L(P),
such that
{} ×E∩L(P)⊂
n
i=
Uui∩
{} ×E.
LetU=ni=Uui. ThenUis a bounded open subset of [, ]×E. From (.), we have
∂[,]×EU∩L(P)⊂ n
i=i
(∂[,]×EUui)∩L(P)
=∅. (.)
Thus,
∂[,]×EU∩L(P) =∅.
From (.) and (.), we have
∂[,]×EU∩S=∅.
Then from Lemma ., we have
degI–T(λ,·),U(),θ=degI–T(λ,·),U(),θ. (.)
SinceU() =∅, then
SinceFixT⊂B(θ,R), then from (.) we have
degI–T(λ,·),U(),θ= –, (.) which contradicts to (.) and (.). Therefore, there must exist (,u)∈({} ×E)∩L(P) such thatDuis bounded.
Step . Obviously, the projection of Duis a interval, denote it by [,λ∗], thenλ∗≤.
ThenDuis a bounded connected component of ([,λ∗]×E). Taker=β(bα–a)φ(λmδ) + , let
Y=
{} ×E∩M[r, +∞), Y=
[, ]×E∩M(r), Y∗=[, ]×E∩M[r, +∞).
Obviously,Du∩(Y∪Y)=∅. For eachp∈Du∩(Y∪Y), denote byE(p) the connected component of the metric spaceDu∩Y∗, which passes the pointp. Now we shall prove
that there must exist ap∈Du∩(Y∪Y) such thatE(p) is an unbounded connected component of the metric spaceDu∩Y∗. On the contrary, assume thatE(p) is bounded
for eachp∈Du∩(Y∪Y). Then, for eachp∈Du∩(Y∪Y), in the same way as in the construction ofU∗in (.) we can show that there exists a neighborhoodU∗(p) ofE(p) in Y∗such that
∂Y∗U∗(p)∩Du=∅. (.)
Obviously, the set of{U∗(p)∩(Y∪Y)|p∈Du∩(Y∪Y)}is an open cover of the set Du∩(Y∪Y) andDu∩(Y∪Y) is a compact set. Thus, there exist finite subsets of
{U∗(p)∩(Y∪Y)|p∈Du∩(Y∪Y)}, say
U∗(p)∩(Y∪Y),U∗(p)∩(Y∪Y), . . . ,U∗(pn)∩(Y∪Y), which is also an open cover ofDu∩(Y∪Y), that is,
n
i=
U∗(pi)∩(Y∪Y)
⊃Du∩(Y∪Y). (.)
LetU=
n
i=U∗(pi), thenUis a bounded open subset ofY∗. Since
∂Y∗U⊂
n
i=
∂Y∗U(pi),
then by (.) we have
∂Y∗U∩Du=∅. (.)
Let
W=
[, ]×E∩M[,r)
andW= ([, ]×E)\Cl[,]×EW. It is easy to see that
∂[,]×EW⊂∂Y∗U∪
M(r)∩
[, ]×E\U∩(Y∪Y)
.
From (.) and (.), we see that∂[,]×EW∩Du=∅. Obviously,W∩Du=∅andW∩ W=∅. Note the unboundedness ofDu, thenW∩Du=∅. Now we haveDu= (W∩Du)∪
(W∩Du), which is a contradiction of the connectedness ofDu. Therefore, there must exist
p∈Du∩(Y∪Y) such thatE(p) is an unbounded connected component ofDu∩Y∗.
Step . Sinceu=T(λ,u)∈Qλfor each (λ,u)∈E(p), we have
u≥αu–βφ–(λmδ)e(t) >θ.
So,u=A(λ,u) for each (λ,u)∈E(p). This impliesE(p)⊂L(P) is an unbounded subset. LetD∗be the connected component ofL(P) containingE(p). Obviously, there existsλ∗> such thatProjD∗⊃(,λ∗]. AsD∗is unbounded, we easily see that
lim
λ→+,(λ,u)∈D∗u= +∞.
The proof is complete.
Corollary . Let(A)to(A)hold.Then there existsλ∗> such that problem(.)has a positive solution uλfor <λ<λ∗withuλ → ∞asλ→.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
Acknowledgements
This paper is supported by Innovation Project of Jiangsu Province postgraduate training project (CXLX12_0979).
Received: 26 September 2012 Accepted: 8 March 2013 Published: 22 April 2013 References
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doi:10.1186/1687-2770-2013-100
Cite this article as:Xian and Xunxia:Structure of positive solution sets of differential boundary value problems.
